Name:

Directions: Show your work! Answers without justification will likely result in few points. Your written work also allows me the option of giving you partial credit in the event of an incorrect final answer (but good reasoning). Indicate clearly your answer to each problem (e.g., put a box around your answer). Good luck!

Problem 1 (10 pts). Below are some terms or properties discussed in class and in this chapter; following those are several possible sentence completions. Choose the most appropriate completion for each term and place its letter in the box next to the term.

1. A circuit
2. A spanning tree of a graph
3. A Hamiltonian circuit
4. A connected graph
5. An Eulerian path

   Possible definitions:

   1. begins at some vertex, goes through every other vertex exactly once, and returns to the starting vertex.
   2. includes each edge of a graph exactly once.
   3. is a graph for which a number is assigned to each edge of the graph.
   4. has a path connecting any two vertices.
   5. has the property that the vertices of the original graph are contained in it.
   6. is a path that begins and ends on the same vertex.

Problem 2 (5 pts). What is Euler's Theorem? How is it connected to the city of Königsberg?

Euler's Theorem tells us the conditions under which an Eulerian path or Eulerian circuit exists in a connected graph (circuit: all even vertices; path: all even, or a single pair of odd vertices). Euler's research into graph theory began with his attempt to resolve the ``Bridges of Königsberg'' problem - the citizens there played a game in which they tried to cross each bridge exactly once before crossing any single bridge twice.

Problem 3 (10 pts). Give examples of the following:

1. A graph with seven vertices which admits an Eulerian circuit.

    

2. A graph with five vertices which admits an Eulerian path, but not an Eulerian circuit, and includes a loop.

    

Problem 4 (10 pts). For the following two graphs, determine whether there is an Eulerian path or circuit (all intersections and corners represent vertices). If so, neatly draw the path or circuit on the graph, indicating clearly the starting vertex and the order in which the edges are traversed.

 

There is a path (not a circuit, as there is a pair of odd vertices) for the graph at left; and there is a circuit (all vertices even) for the graph at right.

Problem 5 (10 pts). Give an Eulerization of the following graph:

 

At left is the original graph, and at right one Eulerization. We circle the odd vertices, then connect pairs of odd vertices.

Problem 6 (10 pts). For the following weighted graph use (i) the Nearest Neighbor Algorithm (starting from vertex A) and (ii) the Greedy Algorithm to find the approximate solution to the traveling salesman problem. On the two graphs at bottom, draw the Nearest Neighbor solution on the left, and the greedy solution on the right, and give the total weights.

 

The total weights:

• Nearest Neighbor: 45
• Greedy: 47

Problem 7 (10 pts). A school bus driver must begin at teh school, then pick up children at four different bus stops: A, B, C, D. The road mileage between the stops is shown below:

 

Draw the corresponding weighted graph, and find the length of the Hamiltonian circuit formed using the greedy algorithm.

 

The length of the Hamiltonian circuit is 30.

Problem 8 (10 pts). Use Prim's algorithm to find a minimal spanning tree for the given weighted graph and give the weight of this tree. Does it matter which vertex you start from?

 

No it doesn't matter which vertex you start from: in this graph there are no ties, so there is a unique minimal spanning tree.

Problem 9 (10 pts). There are 4 cities, which may be represented on the map by coordinates of a rectangular coordinate system as follows: A at (0,0), B at (0,4), C at (3,0), and D at (4,5). Direct routes exist between each pair of cities.

1. (4 pts) Using the distance formula for the distance between cities at and , draw a weighted graph for this situation.

    

2. (3 pts) Use any methods at your disposal to find a Hamiltonian circuit of least total weight linking the four cities, and give the weight of this circuit. Explain how you know that your answer is the circuit of least weight!

    

   Total weight: 16.22. I know that this is the minimum because it is the smallest of the three possible circuits!

3. (3 pts) What is the total weight of the minimal spanning tree for this graph?

    

   Total weight: 11.12

Problem 10 (5 pts). Give a practical example of a real-world application (other than an example from this test) which can be described well as a traveling salesman problem. Use complete sentences, and the vocabulary of this chapter in your description.

There were several mentioned in the book, p. 383.

Problem 11 (10 pts). A census bureau worker was required to visit all homes in the neighborhood whose streets are pictured below:

 

The streets on the outside of the neighborhood have houses on only one side, and the interior streets have houses on both sides. The worker plans to park her car at one of the street intersections and walk down each of the interior streets twice (once for each side) and each of the outside streets once.

1. Draw a graph corresponding to this situation,

    

   and

2. Determine whether or not the graph has an Eulerian circuit, and, if so, give one. If it does not, explain why not.

   There is indeed an Eulerian circuit, as each vertex is even:

    

• About this document ...